# Inference in Bayesian networks with hidden variables

Suppose I have the Bayesian network in the figure and the corresponding conditional probability table for each node, where A and B are the hidden variables, and C and D are the observed variables. What probabilistic inference algorithm can I use to get all the conditional probabilities in Table - 1? can I use likelihood weighting sampling inference algorithm ? If the network becomes the bottom one, is the likelihood weighting sampling inference algorithm appropriate?

Well I don't think sampling is needed here (unless I misunderstand your question / diagram). I believe what is intended is to expand the probabilities using something like the product rule, so that:

\begin{align} P(c1,d1\mid a1,b1) &= P(d1 \mid c1, a1, b1)\cdot P(c1\mid a1,b1) \\ &=P(d1 \mid c1)\cdot P(c1\mid a1,b1) \end{align}

and you have $$P(d1 \mid c1)$$ and $$P(c1\mid a1,b1)$$ in the tables already so you multiply them together.

I assume this is a homework question so I wont give you all the reasons why I did what I did but I will leave with somethings to think about:

(1) How did I do that probability product rule? i.e. How does the probability product rule work.

(2) Why did I expand the product rule with respect c1 and not d1?

(3) Why did $$a1$$ and $$b1$$ disappear in the probability: $$P(d1 \mid c1, a1, b1)$$

Good luck!

• Awesome, I reviewed some books on probabilistic derivation and got the reason for you doing this, thank you!
– LYU
May 28, 2023 at 18:04
• If I stick with the likelihood weighting sampling inference algorithm, would that also be one of the right things to do?
– LYU
May 28, 2023 at 18:10
• Sampling algorithms normally means "using a computer for simulation" but from the information in the question + structure of it I assume it was setup so as to be able to be done without a computer. But yes, you could keep sampling random points according to an algorithm that follows those conditional probability distributions and eventually you should converge to within minimal error of the final solution. Also, if you like my answer please make sure to upvote + accept it :) I sometimes need the reputation on this forum to ask my own bountied questions haha. May 29, 2023 at 1:42
• Actually, what I want to ask is, if the bayesian network contains hidden variables, such as A and B in the figure, is it unreasonable for me to ask questions such as P(c1,d1|a1,b1) and P(c1,d1,e1|a1,b1)? Since A and B are hidden variables, shouldn't A and B be used as evidence?
– LYU
May 29, 2023 at 8:53
• A Bayesian network just gives the assumed information flow from one variable to another. At any point in time it is therefore not unreasonable to query the graph as long as the conditional probability is mapped onto the graph. In real life we may have information on all the variables, or only some of them, and it may be possible to reach information from one node to another, or maybe not. It depends on how you have structured your network May 29, 2023 at 10:16